Problem: For how many different digits $n$ is the two-digit number $\underline{6}\underline{n}$ divisible by $n$? (The expression $\underline{6}\underline{n}$ should be interpreted as a two-digit integer with tens digit 6 and units digit $n$, not as 6 times $n$.)
Solution: Casework is unavoidable. However, we can tell that 1, 2, 3, 5, and 6 will work because of their respective divisibility rules. (We know that 1 works because every integer is divisible by 1. We know that 2 works because any number with an even units digit is divisible by 2. We know that 3 works because the sum of the digits of $63$ is $6 + 3 = 9$, and 9 is divisible by 3. We know that 5 works because any number with a units digit of 5 is divisible by 5. We know that 6 works because $66$ is divisible by 2, since the units digit is even, and by 3, since $6 + 6 = 12$, and 12 is divisible by 3.) The remaining digits are 0, 4, 7, 8, and 9. No dividing by 0! So, we can't have $n=0$. Testing $n = 4$, we see that 64 is indeed divisible by 4. Testing $n = 7$, we see by dividing that 67 is not divisible by 7. Testing $n = 8$, we see by dividing that 68 is not divisible by 8. Testing $n = 9$, we see using the divisibility rule for 9 that 69 is not divisible by 9 (since the sum of its digits is $6+9=15$ is not divisible by 9). Thus, the answer is 1,2,3,4,5,6, which is $\boxed{6}$ digits.